The matrix for rotating an ordinary vector by ϕaround the z-axis is 𝐑(ϕ) ≡( cosϕ -sinϕ

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The matrix for rotating an ordinary vector by ϕaround the...

The matrix for rotating an ordinary vector by $\phi$ around the $z$-axis is $$ \mathbf{R}(\phi) \equiv\left(\begin{array}{ccc} \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{array}\right) $$ By considering the form taken by $\mathbf{R}$ for infinitesimal $\phi$ calculate from $\mathbf{R}$ the matrix $\mathcal{J}_{z}$ that appears in $\mathbf{R}(\phi)=\exp \left(-\mathrm{i} \mathcal{J}_{z} \phi\right)$. Introduce new coordinates $u_{1} \equiv(-x+\mathrm{i} y) / \sqrt{2}, u_{2}=z$ and $u_{3} \equiv(x+\mathrm{i} y) / \sqrt{2}$. Write down the matrix $\mathbf{M}$ that appears in $\mathbf{u}=\mathbf{M} \cdot \mathbf{x}$ [where $\mathbf{x} \equiv(x, y, z)]$ and show that it is unitary. Then show that $$ \mathcal{J}_{z}^{\prime} \equiv \mathbf{M} \cdot \mathcal{J}_{z} \cdot \mathbf{M}^{\dagger} $$ is identical with $S_{z}$ in the set of spin-one Pauli analogues $$ S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right), \quad S_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & -\mathrm{i} & 0 \\ \mathrm{i} & 0 & -\mathrm{i} \\ 0 & \mathrm{i} & 0 \end{array}\right), \quad S_{z}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array}\right) $$

The matrix for rotating an ordinary vector by $\phi$ around the $z$-axis is
$$
\mathbf{R}(\phi) \equiv\left(\begin{array}{ccc}
\cos \phi & -\sin \phi & 0 \\
\sin \phi & \cos \phi & 0 \\
0 & 0 & 1
\end{array}\right)
$$
By considering the form taken by $\mathbf{R}$ for infinitesimal $\phi$ calculate from $\mathbf{R}$ the matrix $\mathcal{J}_{z}$ that appears in $\mathbf{R}(\phi)=\exp \left(-\mathrm{i} \mathcal{J}_{z} \phi\right)$. Introduce new coordinates $u_{1} \equiv(-x+\mathrm{i} y) / \sqrt{2}, u_{2}=z$ and $u_{3} \equiv(x+\mathrm{i} y) / \sqrt{2}$. Write down the matrix $\mathbf{M}$ that appears in $\mathbf{u}=\mathbf{M} \cdot \mathbf{x}$ [where $\mathbf{x} \equiv(x, y, z)]$ and show that it is unitary. Then show that
$$
\mathcal{J}_{z}^{\prime} \equiv \mathbf{M} \cdot \mathcal{J}_{z} \cdot \mathbf{M}^{\dagger}
$$
is identical with $S_{z}$ in the set of spin-one Pauli analogues
$$
S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{array}\right), \quad S_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}
0 & -\mathrm{i} & 0 \\
\mathrm{i} & 0 & -\mathrm{i} \\
0 & \mathrm{i} & 0
\end{array}\right), \quad S_{z}=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1
\end{array}\right)
$$

Key Concepts

Unitary Transformations

Unitary transformations are changes of basis that preserve the inner product in complex vector spaces. These transformations are crucial in quantum mechanics and other fields because they maintain the norm of state vectors. Demonstrating that a transformation matrix is unitary confirms that it conserves probabilities or lengths in the transformed space.

Spin-1 Representations and Angular Momentum Operators

Spin-1 representations provide a way to express angular momentum operators for systems with spin equal to one. The matrices corresponding to S_x, S_y, and S_z, sometimes called the 'spin-one Pauli analogues,' obey specific commutation relations and are used to model the behavior of systems with intrinsic angular momentum. Relating these representations to the generators of spatial rotations bridges classical rotation and quantum spin.

Change of Basis

Changing the basis in a vector space involves representing the vectors and operators in a new coordinate system. This is often done to simplify the form of the matrices involved, such as diagonalizing a generator. In the context of rotations, a proper change of basis can reveal the underlying structure by transforming the matrix representations accordingly.

Exponential Map

The exponential map is a mathematical tool that relates elements of a Lie algebra to elements of its corresponding Lie group. By exponentiating a small generator matrix multiplied by the rotation angle, one recovers the finite rotation matrix. This process connects linear, infinitesimal transformations with finite, nonlinear operations.

Generators of Rotations

Generators are matrices that, when exponentiated, yield the rotation matrices. They capture the essence of how an object rotates in a particular direction (for example, around the z-axis through J_z). These generators form the basis of the Lie algebra related to the rotation group and are central to understanding the continuous symmetries in physics.

Infinitesimal Rotations and Lie Algebras

An infinitesimal rotation is one performed by an exceedingly small angle. In this limit, the rotation matrix can be approximated linearly, and the change is generated by a matrix (the generator) associated with the Lie algebra of the rotation group. This approach is essential for connecting finite rotations with their underlying algebraic structure.

Rotation Matrices

Rotation matrices are used to describe the rotation of objects in space. In three-dimensional space, these matrices, like the one about the z-axis, act on vector components to yield new coordinates after rotation. They belong to the rotation group SO(3), which is fundamental in classical mechanics and geometry.

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